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Measure, Topology, and Fractal Geometry : Undergraduate Texts in Mathematics - Gerald A. Edgar

Measure, Topology, and Fractal Geometry

Undergraduate Texts in Mathematics


Published: 26th November 2007
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From reviews of the first edition:

"In the world of mathematics, the 1980's might well be described as the "decade of the fractal." Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology...the book also contains many good illustrations of fractals (including 16 color plates)."

Mathematics Teaching

"The book can be recommended to students who seriously want to know about the mathematical foundation of fractals, and to lecturers who want to illustrate a standard course in metric topology by interesting examples."

Christoph Bandt, Mathematical Reviews

..".not only intended to fit mathematics students who wish to learn fractal geometry from its beginning but also students in computer science who are interested in the subject. Especially, for the last students the author gives the required topics from metric topology and measure theory on an elementary level. The book is written in a very clear style and contains a lot of exercises which should be worked out."

H.Haase, Zentralblatt

About the second edition: Changes throughout the text, taking into account developments in the subject matter since 1990; Major changes in chapter 6. Since 1990 it has become clear that there are two notions of dimension that play complementary roles, so the emphasis on Hausdorff dimension will be replaced by the two: Hausdorff dimension and packing dimension. 6.1 will remain, but a new section on packing dimension will follow it, then the old sections 6.2--6.4 will be re-written to show both types of dimension; Substantial change in chapter 7: new examples along with recent developments; Sections rewritten to be made clearer and more focused.

From the reviews of the second edition:

"As a non-specialist, I found this book very helpful. It gave me a better understanding of the nature of fractals, and of the technical issues involved in the theory. I think it will be valuable as a textbook for undergraduate students in mathematics, and also for researchers wanting to learn fractal geometry from scratch. The material is well-organized and the proofs are clear; the abundance of examples is an asset for a book on measure theory and topology." (Fabio Mainardi, MathDL, February, 2008)

"This is the second edition of a well-known textbook in the field ... . The book may serve as a textbook for a one-semester (advanced) undergraduate course in mathematics. ... the book is also suitable for readers interested in theoretical fractal geometry coming from other disciplines (e.g. physics, computer sciences) with a basic knowledge of mathematics. The presentation of the material is appealing ... and the style is clear and motivating. ... the book will remain as a standard reference in the field." (Jose-Manuel Rey, Zentralblatt MATH, Vol. 1152, 2009)

Fractal Examplesp. 1
The Triadic Cantor Dustp. 1
The Sierpinski Gasketp. 7
A Space of Stringsp. 11
Turtle Graphicsp. 14
Sets Defined Recursivelyp. 18
Number Systemsp. 31
Remarksp. 35
Metric Topologyp. 41
Metric Spacep. 41
Metric Structuresp. 48
Separable and Compact Spacesp. 57
Uniform Convergencep. 65
The Hausdorff Metricp. 71
Metrics for Stringsp. 75
Remarksp. 81
Topological Dimensionp. 85
Zero-Dimensional Spacesp. 85
Covering Dimensionp. 91
Two-Dimensional Euclidean Spacep. 99
Inductive Dimensionp. 104
Remarksp. 113
Self-Similarityp. 117
Ratio Listsp. 117
String Modelsp. 122
Graph Self-Similarityp. 125
Remarksp. 133
Measure Theoryp. 137
Lebesgue Measurep. 137
Method Ip. 146
Two-Dimensional Lebesgue Measurep. 152
Metric Outer Measurep. 155
Measures for Stringsp. 159
Remarksp. 162
Fractal Dimensionp. 165
Hausdorff Measurep. 165
Packing Measurep. 169
Examplesp. 177
Self-Similarityp. 185
The Open Set Conditionp. 190
Graph Self-Similarityp. 199
Graph Open Set Conditionp. 205
Other Fractal Dimensionsp. 210
Remarksp. 216
Additional Topicsp. 225
Deconstructionp. 225
Self-Affine Setsp. 229
Self-Conformalp. 234
A Multifractalp. 238
A Superfractalp. 242
Remarksp. 247
Appendixp. 251
Termsp. 251
Notationp. 254
Examplesp. 255
Readingp. 255
Referencesp. 257
Indexp. 261
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780387747484
ISBN-10: 0387747486
Series: Undergraduate Texts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 272
Published: 26th November 2007
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 24.3 x 15.6  x 2.2
Weight (kg): 0.59
Edition Number: 2
Edition Type: Revised